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Question #43657

Difficulty: 8/10 | 🟢 Free

The Question

A stationary atomic nucleus of radius $R$ undergoes a transition to its ground state by emitting a gamma-ray photon of energy $E$ . Assuming the universal nuclear mass density is $\rho_0$ and the speed of light in vacuum is $c$ , which of the following expressions correctly represents the recoil speed $v$ of the nucleus? (Assume the recoil is non-relativistic and $E$ is much less than the rest mass energy of the nucleus).

Options

A.
$\frac{3E}{4\pi R^3 \rho_0 c}$
B.
$\sqrt{\frac{3E}{2\pi R^3 \rho_0}}$
C.
$\frac{3E}{2\pi R^3 \rho_0 c}$
D.
$\frac{E}{\pi R^3 \rho_0 c}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Recoil of Atom|Nuclear Density Invariance|Algebraic Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to link microscopic quantum phenomena (gamma emission) with macroscopic nuclear properties (density and volume) using conservation laws.

The primary trap lies in misapplying energy conservation by equating the photon's energy directly to the classical kinetic energy of the recoiling nucleus ( $E = \frac{1}{2}mv^2$ ), leading to the dimensionally consistent but physically incorrect Option B.

Additionally, it requires the realization that nuclear mass must be derived from the invariant nuclear saturation density ( $\rho_0$ ), penalizing students who rely solely on given mass variables without understanding the geometric and material properties of the nucleus.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Recoil of Atom
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free